= Solution
Very near one planet, that planet dominates both the particle's free-precession coefficient and its forcing coefficient, with the leading terms arranged approximately as $\dot z\simeq iA(z-z_p)-z/\tau$. Since $|A|\tau\gg1$ close to the planet, the late-time forced solution approaches
$$
\boxed{z\simeq z_p}.
$$
The particle's orbit therefore tends toward the planet's eccentricity and apsidal direction. Extremely close to the planet, close encounters, co-orbital dynamics, and individual <mean-motion resonances> invalidate the orbit-averaged linear approximation.
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